1. s& is a Generator on V'

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1 Chin. Atm. of Math. 6B (1) 1985 GENERALIZED SEMIGROUP ON (V, H, a) Y ao Y unlong A b stract Let 7 and H be two Hilbert spaces satisfying the imbedding relation V c.r. Let л/г F >F' be the linear operator determined by a(u, v) = (i/u, v) for u, v V, where a(u, v) is a continuous sesquilinear form on V satisfying. a(u, и)+а и я>с м $for u V and some A В and c > 0. In this paper it is proved that.j/is the generator of an analytic Go-semigroup on V. Furthermore, if Ъ(и, v) is a continuous sesquilinear form o n H x F and 3S: U -> F, the linear operator determined by b(u, v) = (SSu, v) for u, v V, then xf Sl is also the generator of Со-semigroup on F'. Also, similar results are proved on inserted spaces Fe(# > 1) which are determined by the spectrum system of sf, 1. s& is a Generator on V' Let V be Hilbert space, Я be the pivot space and Я = Я ' (Я ' is the dual space of Я ). We assume that V is dense and continuously imbedded in Я, and V Hence, we have the inclusions V < zs = H 'c zv '. (1.1) Denote by»j1r (resp.» H) the norm in V (resp. in Я ) and by (.,. )v (resp. (., -)h) the corresponding scalar product. Let and assume that for some c> 0 and that that a(u, v) = continuous sesquilinear from o n F x F (1.2) a(u, м)>с м[.у, (1.3) a(u,»)=a('o, u), \fu, (1.4) By Lax-Milgram theorem, there exists a unique linear bounded operator V ) with domain D (s/) = F dense in V' which is an isomorphism such a(u, -у), \fu, v V, (1.5) * Manuscript received July 22, * Institute of Mathematics, Fudan University, Shanghai, China.

2 28 CHIN. ANN. OF MATH. Vol. 6 Ser. В where <.,. )>denotes the scalar product between V' and V. is Let linear operator A with domain D (A) be the restriction of jf, that A = jtf\d(a)} where D(A) (1.6) It is well known that A is the generator of an analytic semigroup e~m on H. This result is very restrictive and does not allow us to consider unbounded control problem of practical importance. We are going to show that srf is a generator on F '. Set (и, v)a=a(u, v) for u, v( V. It is obvious, by hypotheses of (1.2), (1.3) and (1.4), th a t(.,. )a is a scalar product of F and it is equivalent to the original scalar product (.,. )v. Hence Va ^ L (F, (.,. )a) is also a Hilbert space and V'a= V ' (F a-the dual space of F a). The dual norm of 1*110 in V a is defined by l/u - mp (I/WI/HI.), v /e n -r, (1.7) «F-{0> which is equivalent to the original dual norm \f\v ' in V. Denote by (.,.)_ the corresponding scalar product in F a. Obviously, T t is a semigroup with generator jf on V ' iff so is T t on F a. From <jf m, v}=a(u, v) = (u,v)a, Mu, -dg F, it follows that Now we can show In fact, for m, dg F T heorem 1. jf is Eiesz map from F 0 to Fo. (1-8) (u, '»)h= ( jf m, -»)_, Mu, v V. (1.9) (jf m, -y)_o= (jf -1jF m, jf -1'y)o = (u, jf -i'r)o = a ( ^ 1'y, u) = stf~xv, u) =<a>, w> (w, ч>)н (jf is Riesz map) (1? is a pivot space). The linear operator jf: V >V' is the generator of cm analytic semigroup (denote by e~m) on V which is strongly V'-continuous in t on the right half plarn Ret>=0 cmd strongly V'-analytic an Retf>0. Moreover where D (jf ) is the domain of s s f n. е~м У 'ad(jz? ) = V Ret>0, (1.10) Proof From (1.1),, (1.4), by Lax-Milgram theorem and (1.9), it follows that jf, which is an operator from the subspace F of Fa to V'a itself, is dense definite, surjective, symmetric (u, jf'y)_a=(jfo), w)_o= (v, u)h= ( u, v) h = { ^ u, v ).a for u, v G D (^sf) ( == F ), and positive definite (jfw, 'w)_o= \u\r > bi\u\2-a, MuGDissf) for some <5i>0 (it follows from H Q.V a). Hence jf is a positive definite self-adjoint operator from V'a to itself.

3 No. 1 Yao, Y. La GENERALIZED SEMIGROUP ON (V, H, a) 29 for By the spectral resolution of a self-adjoint operator, there exists {E (!?,, = 0 the resolution of the identity of ssf on V'a such that 'kde%, (1.11) where the integral is in the strongly sense and r. - D O a O - j / g r, (1.12) From (1.16) below, it is easy to show that for Re Setting B >y 'c : D ( s /ao) for A G (, + ). +eo в - ^ ( П ) e - «(1.13) (1.14) 0, we may show that е~^ (Re tf>0) is an analytic semigroup on V by computation. Let us prove the generator of e~m is exactly sd. First, we have for each - r G = F, 8 ( stfv) -a Г+оо e~m- l 2 1 A2d E}v \ -» 0 as t->0, Re >0 (by Lebesgue theorem) U and for e a c h /G F ' F and each complex sequence {tn} (tn >0 as n H-co and Retf» > 0 ), we have ')Г~'П'ЯИпJ- J' ^ Jim fi-»+oo F+oo = lim 71 >+ J 0 f + oo > lim J 0»-»+oo Д» е~мя 1 Un V d \E tf\u A2 d E }J 2 0 = ^ X2d \ E J \ l a=+oo ( / e F ). (by Fatou theorem) Hence, by the definition of a generator, the generator of q~m is sd. Let us prove (1.10). H (jf» ) = { /G F '; j o+v d ^ / 2-0< + oo (n -1, 2,.«) (1.15) and For e a c h /G F ' and :Re <5>0 we have F + OO Г + 09 X2nd\ Еф- ^ j 12_0 = f + OO <const d je?a/ * a< + o o. А2" 10-«IЧ I E %f 12_e Hence e~mf GD(^з/й) and so «**/GH(^F ) = U ( ^ nx 71=1 Remark 1. then Theorem 1 is still true. If for some A>0 and c>0, a(u,,u)+ Л мjя^с м у, VmG F, (1.3)'

4 30 OHIN. ANN. OF MATH. Vol. 6 Bex. В 2. The Inserted Spaces V? For eaoh 1, let the subspaoe V e of V be with the scalar product Fe = { / 6 r ; J * V +4 S J e< + o o j (2.1) ( /, Exg)-a, ( 2.2 ) Then Ve are Hilbert spaces and Vgt+Ve' for eaoh pair (в, 9') whioh satisfies # > # '> 1. It is dear that V - i V ' and ( /, g)_t (J, g)_a for /, g V '. Without confusion,we may use the same sign E %to represent the operator E x н which is E %restricted on H. Lem m a 1. {E A} is a resolution of the identity on H, Proof We need to show the following 1 E, e ^ ( E ), 2 E %is an orthogonal projection on H; 3 A ^ jjb = $ E je /j,= I? ; 4 J57Aa? a?jд >0 as A >+oo for cb^ H. And!?_<*,=0; 6 E y x E ^ x Ih-^O as A > ja + 0 for a j H and ^u-g ( + ). First we show that jsf(jef) for eaoh (, + o ). Indeed for v ^ V, Г+оо that is З Д М ffib Г+оо == A d l ^ l - ^ Ad Eyv \2- а=\г\н1 15д«я< н for v V. (2.3) For each scgu, there exists a sequenoe {г> } in V whioh satisfies l ^ ж н >0 as n >+oo by V ^ E and hence ron~ \va -»0 by HcpV'a. So E ^ - E ^ c o 1v,a >0 as n-> +oo by E v^pfff 'a). By (2.3), we have \E vn- E flvm\h<\'vn vm\h-^0 asn, m->+oo_ Thus there is y E so that Ц Е ^ у\н~>0 as «, > So y^e^oc. Consequently, IE,/on Е,ук #->0 as «-> Substituting v = vn into (2.3), we have As n->+00, it follows that T n u s ^ G ^ ( H ). Now we show 2. For m, dg F I Ец1)п j» я. Д *<Н я<м я for ж Е#. (Е и, v)h= (sv E ^u, v) _0 = Г Ad(E xu, E^v) (by (1.9)) = (w, J2/I?^)_0==(m, Л «)я.

5 No. 1 Yao, Y. L. GENERALIZED SEMIGROUP ON (V, H, a) 31 Hence, by the lim iting process and 1, is symmetric on H. From the definition of F %, we see >==j5> From above, we have proved that Ец is an orthogonal projection e on H. We omit the proof of 3 5 here. le m m a 2. For each 9 ^ 1 and f V ' we ham P ^ +1d F7A/ 2_1= f+~ M \ E xf\*b. (2.4) Proof If f = F Ng, where g V ', then (2.5) can be obtained by computation. For each / V ', (2.4) may be followed by the limiting process. From Lemma 2, it follows that Ve = {f< V'; н (2.5) ( /, g ) e - l l ~ V d W, З Д я fo r/, g V e. (2.6) L em m a 3. I f f V, then f H iff lim F7a, / h<+ * Note that \F f,f\n ^ \ F*nf\ н for each pair (X, g,) Proof The only if part is obvious. The if part is given as follows. From lim 11?л/ н< + 00>it follows that, \ E, f - E j \ l = \E hf \ % - \ B J \! r * 0 as oo and hence there exists exactly only one element oo H such that -Ё?л/ \n ^0 as X >+oo_ Thus \Fxf a? y/-»0 and so f= oo H. ({Ft} is a resolution of identity on V, hence \E b f f\y»-*q as l->+oo fo r/, g V '). We have (by (1.12) and (2.1)) F = F ia n d (и, v) =(u, v)x for u, v V \ (2.7) F '= F - i and ( /, sof'= ( /, g)-i fo r/, g V \ (2.8) E = V о and (oo, у )н ~ (со, y )0 for to, y JE, (2.9) D ( A ) - V a. The proof of (2.9). We have For oo JE, E = {f V '-, lim J7A/ i < + } (by Lemma 3) A-»+ ={ / G F V j o+ d F7A/. < + o o j / G F '; Xd F/A/ 2i< + o o = Fo (by Lemma 2)c Mirг= Р й. В Д = Г ~ ы м М -: and hence for so, у H, (oo, y)u=*(oo, y )o. The proof of (2.10), (2.Ю) i= M o, (2.11)

6 32 CHIN. ANN. OF MATH. Vol. 6 Ser. В ( A) = 4 / F ; / <s F aiid / Я} f f+oo /*+co "j /G F '; AdjjE7A/ s < + and djjs7a^ / l < + {/er, +"w ^/,< + oo (by :Fa % 4 \F,f\\),. 4 - ( Я ) KdHx. (2.12) Proof We have (Я ) Г~ r+ ' k d F ^ j t f and the domain of чf+' < 4 MF% is Я ( (Я ) j*~ Ы Яя) = { / G F 'i ~ X2d F J 1 < + ooj- V a= Я(Л.). Hence (2.12) is valid. R em ark. Similarly, we have 1. Яя ув is a resolusion of identity on Ve; 2. If Я ( ^ 0) = x /_1F e, then C+oo ^ = ( F e ) j o ЯйЯя and is a generator of the analytic semigroup e~^ei on F e and f+o e * - ( F e)j e-a,d M 3. e~^e,t\ys=e~^st for each pair (#, Theorem % 3. Some Properties of e~* For each # > l, e~m given by (1.14) has the following properties'. I е For each t: Retf>0, the rcmge~md V e (wad e_^ig=f (F /, Fe); 2 For each / G F ; and each s G (0, яг/2) lim ^ +e> /V ^ / Fe = 0; t->0 (Ret>0, argtf < jf/ 2 s) 3 For each /G F ', e~m f is VQ-cmalytic function of t on Re t >0. Proof From (1.14), 1 3 can be obtained by computation. 4. Perturbation Results Suppose b{u, v) =continuous sesquilinear form on F «x F for some # > 1. (4,1)

7 No. 1 Yao, Y. L. GTNERALIZED SEMIGROUP ON (V, H, a) 33 Set <%: b(u, r)= < ^w, for u, v V. By Lax-Milgramtheorem, & Q JP (ye, V '), and we haye the following perturbation results: Theorems. Under the assumptions of (1.1, 2, 3, 4), (4.1) and 1 > 0 > 1, sv 3 is a generator of a G0-semigroup Ut on V ' and it has the following properties 1 rang Utc V e fo r >0; ' 2 lp+'>'*utf O a 0, +oo),v e )fo r f V e -, 3 Ue\re is a O0-8emigroup o n V e with generator A 9= (sv Л-38) \dcao» where B {A e) = I f 9 = 0, then Ut \u is a O0-semigroup on H with generator A0= (sv-vss) u(ao)у where B {A 0) = (j/ + ^ ) _1JI. Proof Let us consider the integral equation on V в в (0 -в -л/ + ( Г в) * в - )( - - ^,)в(*)(й, for >0, (4.2) U rf 99 where / 6 V and the integral (V$) is well-defined Boohner integral in Vo- Set «= (1 + 0 )/2 (< 1 ). Theorem 2 shows that \e ^ S cnfs)<const/^ for 6(0, *), x-an given arbitrary positive number and e~mf ^ O { { 0, +oo), V f). Hence I e-*(t~s) ( ^(Гв) <const ( - s)- and е~,а(3>~й'>38 is strongly continuous on > s> 0. So there is an unique solution»(0 GO((0, +oo), V 9)f)B i(0, +oo; V e) (4.3) of (4.2) whioh may be represented by. and «( O - e ^ Z + O ^ ^ <?(*-«>" * fds for >0 (4.4) G (t)^ G o (t)/tae ^ ( 7 e ) for >0, (4.5) here G0(t) JP(Ve) and it is strongly F e-continuous on 6[0, +oo), and?( ) is the unique solution of Set Gt(i5) = e- ^ ( _ ^ ) + (F 9) % - ^ - s4 - ^ ) G i(s)ds, for >0. (4.6) Hence from (4.3) we have «(O-ff./for/Gr. (4.7) ff(«e- ^ + ( F e)j* <?( -«> " ds} where the integral is in the strong sense.. Using Theorem 2, from (4.4, 5, 6) it follows 1, 2 of Theorem 3. From (4.2) and (4.7), by V o^w ' we have x{t)==utf ^ e - Mf + { V f) ^ e - ^ - s\ - ^ ) U sfd s for >0. So the F'-continuity of Utf on the left on [0, + oo) follows from the strong F'-oontinuity of e~m on* 6 [0, + oo) and the faot that - m sf e L i(0, «V ') (by 2 of Theorem 3)

8 CHIN. ANN. OP MATH. Vol. 6 Ser. В for i>0. For the semigroup property of Ut we can easily show, similar to [1] p. 277, (Ut+s- U tua) f= (Ve) Г e ~ ^ \ (U*+t- U J J.) f dt ( for t, s> 0. Hence, by the uniqueness of the solution, Similary Ut+af^UtUsf for t, s>0 a n d /G F '. and Q-(t) 6 ii( 0, + o o ; Ye), and hence Gf(t) is strongly F'-continuous on 2 [0, + o o ). Denote the generator of Ut by J f. From (4.4, 7) f o r / F F ' - lim J b L - L - V - lim e~* -Zl.f+V'-Jsm A f*gf(t-s)e-*sfds f-»+0 t $-H-0 1 f-*+o t. - - s / f + G ( p ) f s / f + ( - & ) f (by (4.6)). Hence Bet We have 1*41 J f i D jf J?. Ъц(и, v) }Jb(u, «)я+«(м, v)+ b(u, v) for u, v(zv ^ ~ % 4 \E,u \% j o ( )ved E,.u ( d \Е-,ц ) (by Holder inequality) (4.8) There is a sufficiently large /j,0> 0 suoh that /jo> iiq implies cya/ 2 С2/у1+в+ /л к= д(у)> 0 for y> 0 and fjb J/f is one to one (by the property of generator). have 15#.(**, м ) > /л м н + а («, и )\-\Ъ (и, м) >/а м н+с[м Я м я const м в* w y Hence for some c2>0, we > c\u\% /2+ \u\% g(\u\^/\u\v)> c\u\^/2 for mgf ' and hence, by Lax-Milgram theorem /л,+ з/+ & is surjective for ^> /л0>0. From (4.8), we have fju zd jjj-f-sy + where /ju+<%?+& is surjeotive and /л J f is One to one, so that is Reference [ 1 ] Cartain, B. P. & Pritchard, A. T., Lecture Notes in Control and Information Sciences, Vol. 8, Springer-Verlag, 1978.

ITERATION OF ANALYTIC NORMAL FUNCTIONS OF MATRICES

Ohin. Ann. of Math. 6B (1) 1985 ITERATION OF ANALYTIC NORMAL FUNCTIONS OF MATRICES Tao Zhigttang Abstract In this paper,, the author proves that the classical theorem of Wolff in the theory of complex